Shape optimization, PDE, and Nonlinear Analysis

7 oct. 2026, 09:00 → 9 oct. 2026, 15:50 Europe/Paris

Conference in shape optimization, PDE, and Nonlinear Analysis, which will take place in Toulon (France), from the 7th to the 9th of October, 2026.

 

Speakers:

• Pierre Bousquet (Toulouse)
• Blanche Buet (Paris-Saclay)
• Nicolas Clozeau (Toulon)
• Marc Dambrine (Pau)
• Mathis Dauchy (Nice)
• Ilias Ftouhi (Nîmes)
• Xavier Lamy (Toulouse)
• Théo Lavier (Toulon)
• Benjamin Lledos (Nîmes)
• Alba Lia Masiello (Naples)
• Léa Mazzouza (Marseille)
• Simona Rota-Nodari (Nice)
• Giacomo Vizzari (Toulon)

 

Organizers: Thierry Champion, Reza Pakzad, Enea Parini, Rémy Rodiac

Map of La Garde campus

mer. 7 octobre

14:00 → 14:50 Convergence of Ambrosio-Tortorelli functionals on varying surfaces

We study Ambrosio-Tortorelli approximations of the Mumford-Shah functional on varying hypersurfaces, formulated in terms of Sobolev functions on rectifiable currents. We prove a compactness result for bounded-energy sequences and establish a lower bound for a generalized Mumford-Shah functional on the limiting current. In dimension two, we obtain the optimal lower bound and construct recovery sequences for a broad class of configurations.

Orateur: Benjamin Lledos

15:00 → 15:50 About the maximization of the gradient of the torsion function

In this talk, we address the problem of maximizing the supremum norm of the gradient of the torsion function over planar convex domains with fixed measure (or perimeter). We prove that an optimal shape exists, its boundary possesses $C^1$ regularity, and it features at least one straight segment. The proofs leverage probabilistic methods along with a novel boundary Harnack principle for the torsion function. The talk is based on a joint work with Krzysztof Burdzy (University of Washington) and Phanuel Mariano (Union College).

Orateur: Ilias Ftouhi

16:00 → 16:25 Coffee break

16:30 → 17:20 Sharp and quantitative bounds for torsional rigidity with Dirichlet and Robin boundary conditions

Torsional rigidity is a classical quantity associated with the Poisson problem with Dirichlet boundary conditions and is closely related to the geometry of the underlying domain. In this talk we consider open, bounded and convex sets in R^n and present sharp geometric inequalities relating torsional rigidity to basic geometric quantities such as the perimeter, the measure of the set, and the inradius. We also discuss quantitative versions of these inequalities, where the deficit from the optimal constant provides information on the geometry of the optimal sequences. Finally, we address extensions of these results to torsional rigidity with Robin boundary conditions, focusing on the case of positive Robin parameter and highlighting the main similarities and differences with respect to the classical Dirichlet setting.

Orateur: Alba Lia Masiello

jeu. 8 octobre

09:00 → 09:50 Flagfolds: multi-dimensional varifolds to handle discrete surfaces

We propose a natural framework for the study of surfaces and their different discretizations based on varifolds. Varifolds have been introduced by Almgren to carry out the study of minimal surfaces. Though mainly used in the context of rectifiable sets, they turn out to be well suited to the study of discrete type objects as well. While the structure of varifold is flexible enough to adapt to both regular and discrete objects, it allows to define variational notions of mean curvature and second fundamental form based on the divergence theorem.
Thanks to a regularization of these weak formulations, we propose a notion of discrete curvature (actually a family of discrete curvatures associated with a regularization scale) relying only on the varifold structure. Though flexible, varifolds require the knowledge of the dimension of the shape to be considered. We then embed all d-dimensional Grassmannians into symmetric positive semi definite matrices with trace 1, that we endow with a distance coinciding with the Riemannian one in each Grassmannians.
Building upon the aforementioned embedding of Grassmannians, we propose a generalization of varifolds, that we call flagfolds, in order to model multi-dimensional shapes. The notion of first variation extends to such flagfolds and we are investigating whether some form of Allard's rectifiability theorem extends as well.

Orateur: Blanche Buet

10:00 → 10:25 Coffee break

10:30 → 11:20 Quantitative stochastic homogenization of variational models arising in fracture mechanics

I will present a recent quantitative result concerning the stochastic homogenization of the so-called Griffith type model arising in fracture mechanics: given a body $\Omega\subset \mathbb{R}^3$, the energy $E_\varepsilon(u)$ for deformation $u\in \mathrm{SBV}(\Omega)$ takes the form of

$$E_\varepsilon(u):=\int_{\Omega\backslash S_u} F(\tfrac{\cdot}{\varepsilon},\nabla u)+\int_{S_u}g(\tfrac{\cdot}{\varepsilon})\dd\mathcal{H}^{d-1},$$ where $F$ denotes the stored elastic energy, $g$ the toughness and $\varepsilon\ll 1$ the scale of the microstructure. Since the work of Cagnetti, Dal Maso, Scardia and Zeppieri, the homogenized model has been identified qualitatively by taking the $\Gamma$-limit as $\varepsilon\downarrow 0$ of \eqref{Energy}; and in particular the two main constitutive properties of the system have been derived: the effective stored elastic energy and the effective fracture toughness, both given explicitly by means of cell-formulas. I will explain in this talk how we can derive quantitative estimates for the convergence of the cell-formula for the effective toughness. This is based on a joint work with Julian Fischer and Antonio Agresti.

Orateur: Nicolas Clozeau

11:30 → 12:20 On a quasilinear Schrödinger equation: the small frequency limit

In this talk, I will present some recent results on a quasilinear Schrödinger equation with a power nonlinearity. After showing the uniqueness and the non-degeneracy of the positive radial solution $u_\omega$ for all $\omega>0$, I will describe its asymptotic behavior in the limit $\omega\to 0$. This gives some important information about the orbital stability of $u_\omega$ and the uniqueness of normalized ground states. Joint work with François Genoud.

Orateur: Simona Rota-Nodari

12:30 → 13:50 Lunch

14:00 → 14:30 Minimization of a quasi-linear Schrödinger functional for generalized $\alpha$-non-linearities

In this talk, I will present the minimization of a quasi-linear Schrödinger functional derived from an effective model in quantum physics, with generalized $\alpha$-power non-linearities. We generalize, for $\alpha >0$, the existence result already known for $\alpha = 1$. More precisely, we show the existence of a critical coupling constant such that a minimizer always exists above this threshold, while there are no minimizers below it.

Orateur: Mathis Dauchy

14:40 → 15:10 Energy scaling for von Kármán elastic plates with positional constraints

The studies behind the behavior of elastic plates under different kinds of stress (and the resulting deformations) have often focused on models that study the mid-plane section of the plate, reducing it to a two dimensional problem. One of the most studied models is the von Kármán model, with energy functional
$$ U_h = \int \bigg|\frac{1}{2}\nabla v\, \otimes \nabla v \,+\,\text{sym}\,\nabla w\bigg|^2 + h^2 |\nabla^2 v|^2\,dx $$ dependant on the bending stiffness h of the plate. This talk presents the problem of finding bounds for the scaling of the elastic energy as $h \to 0$ as we limit the bent plate in the 3D space through positional constraints by placing the plate between two obstacles. The question is non trivial because of the interplay of the two expressions that compose the energy: with generic positional constraints we cannot ensure for a 0-energy minimizer and, in fact, looking at parallel results for the Föppl-von Kármán model we can conjecture a scaling of order $h^{\frac{5}{3}}$. We will see how this energy scaling can be attained on the upper bound via an approximation by piecewise affine maps, which is born from the purely geometrical idea of origami maps.

Orateur: Giacomo Vizzari

15:20 → 15:50 Spectral analysis of the Dirac operator in shrinking domains

I will be presenting a theorem on the spectral analysis of the Dirac operator with infinite mass boundary conditions in thin planar domains. The main objective is to understand how the geometry of the domain shrinking influences the energy levels and localization properties of the eigenmodes. Using tools from pseudo-differential calculus and semiclassical analysis, we derive asymptotic expansions for the eigenvalues and eigenfunctions in the thin-domain regime. The results reveal the influence of the geometry in shaping the spectral behavior of the operator in the semiclassical limit.

Orateur: Léa Mazzouza

16:00 → 16:25 Coffee break

16:30 → 17:20 Shape optimization in thermoelasticity

This presentation focuses on shape optimization in thermoelasticity, motivated by the design of engine components subjected to thermal stresses.

First, I will discuss a linear framework in which the objective function—a von Mises norm under volume constraint—is analyzed using shape calculus and an adjoint state. This calculation involves a subtlety related to the boundary condition of the thermal adjoint, which becomes non-trivial as soon as the criterion weights the final time point. The framework is then extended to include a robust criterion, where temperature is modeled by a random field and the criterion becomes an expectation; thanks to the quadratic structure of the functional, this can be calculated exactly using a two-point correlation.

In a second step, the assumption of linearity is abandoned for temperature-dependent material parameters. This introduces reaction terms and tangent tensors into the sensitivity analysis, and, most importantly, a stronger coupling between the two adjoint states, which were previously independent. These new challenges are illustrated by numerical results on two test cases.

Orateur: Marc Dambrine

20:00 → 22:00 Social dinner

ven. 9 octobre

09:00 → 09:50 On the Lavrentiev phenomenon for multidimensional problems in the Calculus of Variations

We present some results on the Lavrentiev phenomenon for some scalar problems in the multidimensional Calculus of Variations. Our main goal is to identify a unifying and natural framework that entitles one to discard this phenomenon and the related Lavrentiev gaps.

Orateur: Pierre Bousquet

10:00 → 10:25 Coffee break

10:30 → 11:00 Conductivity in Random Media: From Nonlinear Diffusion to Dielectric Breakdown

We present work in progress on quantitative homogenization results for viscosity solutions of $\infty$-Laplacian-type equations of the form$$\nabla\!\left(a\!\left(\tfrac{\cdot}{\varepsilon}\right)\vert{}\nabla u_\varepsilon\vert{}^2\right)\cdot \nabla u_\varepsilon = 0,$$subject to Dirichlet boundary conditions in dimensions $d\leq 3$, where the coefficient field $a : \mathbb{R}^d\rightarrow \mathbb{R}$ is random and sampled from a stationary, ergodic, and isotropic probability measure. We approach this problem using a quantitative power-law approximation of the solution $u_\varepsilon$, combined with quantitative homogenization results for nonlinear elliptic equations with $p$-growth for $p>2$.

Orateur: Théo Lavier

11:10 → 12:00 Concentration properties for the Aviles-Giga energy

The Aviles-Giga energy is a phase transition model related to liquid crystals, micromagnetics and elasticity. Sharp interface limits of bounded energy are weak solutions of the 2D eikonal equation: unit vector fields $m$ with zero divergence (in the sense of distributions). Partial information about the limit energy cost of a given solution $m$ is encoded in a family of signed measures called entropy productions. It is conjectured that these measures are concentrated on the 1-rectifiable jump set of $m$, as they do if $m$ has bounded variation (BV). I will present two types of partial results towards this conjecture, from joint works with Elio Marconi, and with Andrew Lorent and Guanying Peng.

Orateur: Xavier Lamy